Problem: You have found the following ages (in years) of 6 sloths. The sloths are randomly selected from the 49 sloths at your local zoo: $ 1,\enspace 22,\enspace 9,\enspace 10,\enspace 11,\enspace 12$ Based on your sample, what is the average age of the sloths? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 49 sloths, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{96.04} + {125.44} + {3.24} + {0.64} + {0.04} + {1.44}} {{6 - 1}} $ $ {s^2} = \dfrac{{226.84}}{{5}} = {45.37\text{ years}^2} $ We can estimate that the average sloth at the zoo is 10.8 years old. There is a variance of 45.37 years $^2$.